Qatar’s monarchy on Wednesday overhauled the country’s cabinet, in a sign that the young emir is attempting to better cope with falling energy prices and intensifying regional conflicts.

The cabinet reshuffle, which includes a new minister of foreign affairs and a minister of state for defense, comes at a time when the energy-rich country and its Persian Gulf neighbors have been hit by collapsing oil prices. Income from oil and gas sales makes up the bulk of their revenue, forcing them to reassess their spending plans at...